Linear Dependence In $\mathbb{R}$ Over $\mathbb{Q}$

Let $V$ be a vector space over any field $F$ .

A collection of vectors $v_1, v_2, \ldots, v_n \in V$ is called dependent if there exist real numbers $a_1, a_2, \ldots, a_n \in \mathbb{R}$ such that $a_1 v_1 + \cdots + a_n v_n = 0$ and at least one of the $a_i$ 's is nonzero. Consequently, a collection of vectors is called independent if it is not dependent.

Note that $\mathbb{R}$ may be thought of as a vector space over the field $\mathbb{Q}$ of rational numbers. With this vector space structure on $\mathbb{R}$ , is the set $\big\{\sqrt{2}, \sqrt{3}, \sqrt{5}\big\} \subset \mathbb{R}$ dependent or independent? What about the set $\left\{\frac{\pi}{4}, \arctan\left(\frac{3}{2} \right), \arctan(2) \right\} \subset \mathbb{R}?$